Papers

Let $(\mathcal{S},\mathcal{Z},\mathcal{V})$ be a premutation triple in an extriangulated category. Set $\mathcal{I}=\mathcal{S}\cap\mathcal{Z}=\mathcal{Z}\cap\mathcal{V}$. Iitsuka introduced premutation triples as a unified framework for various mutation constructions and showed that the quotient category $\mathcal{Z}/[\mathcal{I}]$ admits a canonical pretriangulated structure. In this paper, we prove that the compatibility condition $(\mathrm{MT4})$ is not only sufficient but also necessary for this structure to be triangulated. Therefore, $(\mathrm{MT4})$ precisely characterizes when Iitsuka's canonical pretriangulated structure becomes triangulated. Furthermore, we prove that, for concentric twin cotorsion pairs, the compatibility condition used by Nakaoka in the construction of triangulated quotients is equivalent to $(\mathrm{MT4})$. Finally, we construct an explicit reducible triple which is not a mutation triple. This gives a negative answer to a question posed by Iitsuka.

We develop tools to compute moduli spaces of geometric functorial field theories as mapping spaces of equivariant simplicial presheaves. Given a d-dimensional geometric structure F, presented as a presheaf on the site of smooth families of d-manifolds, we define its Cartesian realization, which is an O(d)-equivariant simplicial presheaf on the site of Cartesian spaces. We use Cartesian realizations to present the moduli space of functorial field theories with geometric structure F as a mapping space between O(d)-equivariant simplicial presheaves. In a companion paper, we use this result to compute the moduli space of smooth one-dimensional oriented Riemannian functorial field theories valued in an arbitrary smooth symmetric monoidal infinity-category.

Cartesian 2-fibrations provide a way to understand indexed categories, but their classical “straightening” construction requires several layers of weak coherence data. This paper develops a homotopical framework that replaces much of this bookkeeping with a fully strict model. By using marked 2-categories to record the cartesian morphisms and 2-cells, we construct a model structure whose fibrant objects are precisely the cartesian 2-fibrations over a fixed 2-category $\mathcal{C}$. We then show that the marked Grothendieck construction identifies these 2-fibrations, up to weak equivalence, with strict 2-functors from $\mathcal{C}$ into $2\mathrm{Cat}$. As an additional contribution, we construct localizations of 2-categories that simultaneously invert selected morphisms and 2-cells.

Let $A$ be an Artin algebra and let $T$ be a $\tau$-tilting right $A$-module. Put $C=A/{\rm Ann}_A(T)$. We prove that finite little finitistic dimension of $C^{\rm op}$, together with a finite range of self-extension vanishing for $T$, forces $T$ to be a classical $1$-tilting module. We also show that if the finitistic conjecture holds for all Artin algebras then the self-orthogonal $\tau$-tilting conjecture holds for all Artin algebras. Moreover, a counterexample to the self-orthogonal $\tau$-tilting conjecture will give a counterexample to the finitistic dimension conjecture. As an application, the self-orthogonal $\tau$-tilting conjecture holds for Artin algebras with radical cube zero.

Dynamical systems theory primarily concerns the study of different forms of invariance, such as invariant sets, densities, measures, and observables. Ergodic systems are a special class of dynamical systems which are measure theoretically irreducible. In spite of this specialized property they dominate most discussions on ergodic theory because of the ergodic decomposition theorem. The viewpoint that any dynamical system is a composite of multiple ergodic components enables us to partition the face space into the basins of the different measures. The coexistence and mutual connections between these various coexisting subsystems are illuminated very effectively using the language of Category theory (CT). Some recent advancements have shown how the essence of measure theoretic dynamical systems can be captured through the formalism of Markov categories. This formalism captures the essential features of invariance and ergodicity through the language of limits and colimits. This article continues that formalism by studying the concept of joins using the language of spans and push-outs. A classical result is re-proven which establishes the connection between joins, ergodicity and mixing. In the process, the categorical language is refined to capture the notion of sub sigma-algebras and their invariance.

In this paper, we study certain categories of labeled finite rooted ordered trees over a fixed set of labels where each label is equipped with an arity: a fixed number of children that the vertex with the given label must have. Equivalently, these are expression trees for operations in a free non-symmetric operad. A morphism between these trees matches a pruning of one tree (a prefix) with an entire subtree of another (a suffix). We characterize such categories, up to isomorphism, in terms of suitable exactness properties. It turns out that these categories exhibit strong algebraic behavior, in the sense that every such category, when appended with a strict initial object, has a particularly nice noetherian form.

We establish local Ocneanu rigidity for separable algebra objects in Hom-finite monoidal categories C over an algebraically closed field k. A separability morphism contracts the first two Hochschild cohomology groups without requiring an abelian ambient category. Separable algebra structures and homomorphisms from separable sources have open algebraic-group orbits, and affine Bezout estimates give effective finiteness results. For Frobenius subalgebras, exchange relations replace the source and embedding data by a single self-dual idempotent. We prove that their separable inner-conjugacy classes are open in the exchange locus, yielding a bound of 2^(dim_k(End_C(X))) for any ambient Frobenius algebra X; connectedness gives the same bound on the actual number of subalgebras. As an application, we obtain an effective form of the Etingof-Walton finiteness theorem: every finite-dimensional semisimple Hopf algebra H over C has fewer than 2^(dim_C(H)) left coideal subalgebras. In the unitary setting, we bound E-compatible intermediates of C*-algebra and von Neumann algebra inclusions up to unitary conjugacy, under finite-index and finite-center hypotheses. For irreducible finite-index subfactors N < M, we improve the 9^[M:N] bound of Bakshi-Das-Liu-Ren to fewer than 2^[M:N] intermediate subfactors.

Originating in ergodic structure theory, nilspace theory is concerned with the study of cubical spaces. A first key result in this theory, the weak structure theorem, is the representation of a concrete fibrant compact cubespace as a tower of its canonical truncations, exhibiting each step over the previous one as a principal bundle by a compact abelian group. We present a categorical proof of the weak structure theorem by systematically interpreting the relevant concepts in an appropriate context.

We prove a duality theorem for lax equalizers of stable $\infty$-categories, and we apply it to Frobenius and Cartier modules on a Noetherian $F$-finite scheme $X$ of characteristic $p$ with a unit dualizing complex. When $X$ has affine diagonal, comparison with the derived categories of coherent modules and passage to homotopy categories recovers Baudin's derived Hom duality. The duality of coherent modules exchanges the standard $t$-structure on one side with the perverse $t$-structure determined by the dualizing complex on the other. In particular, it identifies the abelian category of coherent Cartier modules with a perverse heart of coherent Frobenius modules, and conversely.

We study universal derived equivalences between diagram categories indexed by finite posets. Starting from a construction of Ladkani, we give an intrinsic criterion for determining when a finite poset admits a decomposition to which this construction can be applied. This leads to the notion of an admissible cut, formulated entirely in terms of the order structure of the poset. Our main result therefore provides a method for producing, from a given finite poset admitting such a cut, a new poset that is universally derived equivalent to it. The construction is reversible once the partition is retained, so that the original mixed order relations can be recovered from the transformed poset. As applications, we show that every finite poset of height at most one is universally derived equivalent to its opposite, give a criterion characterizing source-to-sink transformations at minimal elements, and recover the universal derived equivalence of all orientations of a finite tree through sequences of such local transformations. These results are also applied to persistence modules indexed by finite posets and to extensions obtained by adjoining further finite parameters.

This paper introduces a natural fuzzy order on fuzzy numbers that extends the natural orders on real numbers and interval numbers. We investigate its completeness properties and show that the space of uniformly bounded fuzzy numbers is conically complete and conically cocomplete, and that it is complete if and only if the underlying continuous t-norm is the Gödel t-norm. Moreover, it is proved that this space constitutes a \([0,1]\)-enriched domain if and only if the underlying continuous t-norm satisfies the (S) condition. These results provide a foundation for ordering fuzzy numbers.

A categorial approach to the non-metric geometry in Books V and VI of Euclid's Elements is presented. Specifically, we introduce a diagrammatic syntax that can be overlaid immediately on his diagrams, thus bridging intuitive presentation with fidelity to Euclid's arguments. This syntax makes complicated definitions like V.5, and indeed the arguments throughout books V and VI, including arguments about similar figures, intuitively clear. We show in an appendix that this syntax can be used to solve a puzzle regarding ancient mathematics. Finally, we offer evidence that this approach to Euclidean diagrams is rooted in the Aristotelian tradition itself, and that a similar syntax was utilized, in antiquity, for related questions of numeric and proportions. Thus the syntax is plausibly faithful to Euclid's own thought-world, and not an outside-imposition.

Classically, the places of $\mathbb{Q}$ are often regarded as a one-point compactification of $\mathrm{Spec}(\mathbb{Z})$, with the real place corresponding to a formal “prime” added at infinity. Re-examining this picture from a topos-theoretic perspective reveals a subtler geometry: while the non-Archimedean places are identified with singletons indexed by the non-zero prime ideals of $\mathbb{Z}$, the Archimedean place is represented by the space of upper reals $\overleftarrow{[0,1]}$, which may be informally thought of as the unit interval equipped with a non-Hausdorff topology. On a technical level, our analysis brings together geometric logic and descent techniques from topos theory, distinguishing standard descent from lax descent toposes both at the level of sheaves and of the geometric theories they classify. More broadly, this paper brings into conversation two parallel distinctions: on the number-theoretic side, between Archimedean and non-Archimedean phenomena, and on the topos-theoretic side, between standard and lax descent. Looked at from a high level, these perspectives begin to converge on a common theme: how should the connected and the disconnected interact?

Modern science and engineering increasingly rely on time-varying data, yet the mathematical tools used to model temporal phenomena are often developed within separate disciplines, obscuring common principles and limiting the transfer of ideas across fields. This chapter presents the theory of narratives, an abstract framework for time-varying objects of any mathematical kind that supports both theoretical investigations and applications. To illustrate this perspective, the chapter develops three vignettes, each illustrating a different research direction. The first addresses a general concern: What information loss can occur when switching between different representations of temporal data? The second concerns structural and algorithmic approaches: How can we systematically decompose time-varying data into simple pieces and obtain invariants describing its structural complexity? The third is an application to control theory: How can we model multi-agent systems with switching communication topologies? More important than any individual vignette, the central message of this invitation is that a suitable abstract perspective can organize and guide research across remarkably diverse mathematical and scientific domains.

Toward modeling computational concurrency using the language of central idempotents in monoidal categories, we give an explicit construction of the pushout of strict symmetric monoidal categories with strong monoidal functors between them. We show that any braided strong monoidal functor strongly preserves central idempotents, and as a corollary, that the lattice of central idempotents of a symmetric monoidal category is isomorphic as a meet-semilattice to that of its strictification.

This paper gives a categorical interpretation of Baumeler & Wolf's logically consistent non-causal circuits, connecting them to postselected quantum teleportation. Looped feedback is represented by a trace in the category of non-negative matrices, and it is shown that the traced process is stochastic precisely when the induced loop transition matrix has trace $1$ for every external input, a condition shown to be equivalent to a unique fixed point for the loop for each input to a deterministic circuit. A classical non-causal circuit is represented by a measure-and-prepare quantum channel with an internal register utilising a maximally entangled Bell-state with postselection. The main result is that classical logical consistency is equivalent to the postselected Bell outcome having, for loop dimension $d$, probability exactly $1/d^2$ for each classical input distribution. The subsequent normalised conditional output then agrees exactly with the classical categorical trace. This work identifies a class of quantum Bell-postselection constructions whose conditional evolution maintains linear dependence on classical input distributions.

We give a new equivalent formulation of Verdier's octahedral axiom. An initial square in a pre-triangulated category determines an obstruction in a quotient of ordinary Hom groups. Its vanishing is equivalent to the existence of a good completion, and $\mathrm{TR4}$ is equivalent to this vanishing for the squares associated with composable morphisms. As a consequence, if two pre-triangulations have the same underlying additive category and one is triangulated, then the other satisfies $\mathrm{TR4}$ precisely when their relative inverse Heller comparison admits an exact lift along every short exact sequence in the Freyd category. As applications, we prove that the exotic pre-triangulated category of Díaz Cabrera–Muro is in fact a triangulated category over every algebraically closed field. We construct scalar families for type-$A_{3a-1}$ preprojective algebras over arbitrary fields and for a range of Dynkin preprojective algebras in characteristic two; in each family the zero parameter is the unique triangulated member. The local equations also yield separable descent, including canonical triangulations for non-modular equivariantizations and for the Markman–Mehrotra K3 deformation categories.

Let $A$ be a split finite-dimensional algebra over a field whose radical $J$ satisfies $J^3=0$, and let $s$ be the number of isomorphism classes of simple $A$-modules. We prove that a non-projective module $M$ with ${\rm Ext}_A^i(M,A)=0$ for all $i>0$ has a non-zero self-extension in some degree between $1$ and $3s+1$. In particular, $A$ satisfies the Auslander–Reiten conjecture, which asserts that every self-orthogonal generator is projective. As a consequence, every finite-dimensional algebra over an algebraically closed field with radical cube zero satisfies the Auslander-Reiten conjecture.

This is a paper about contraherent cosheaves on non-semi-separated schemes. We prove two theorems, a negative one and a positive one. On the negative side, let $X$ be a quasi-compact, quasi-separated scheme that is not semi-separated. We present a locally cotorsion contraherent cosheaf on $X$ that does not have an admissible monomorphism into any locally injective locally contraherent cosheaf. The construction and proof follow the arguments of Slavik and Stovicek in arXiv:1902.05740. On the positive side, let $X$ be a Noetherian scheme of finite Krull dimension. We prove that every $\mathbf W$-locally contraherent cosheaf on $X$ has an admissible monomorphism into a locally cotorsion $\mathbf W$-locally contraherent cosheaf. Moreover, the cokernel is a flat contraherent cosheaf. The proof is based on the theorem of Raynaud-Gruson about the projective dimensions of flat modules and Enochs' classification of flat cotorsion modules.

For a crossed module $\chi: G \to H$, we introduce the notion of $\chi$-crossed braided (resp. ribbon) categories, where the categories are graded by group $G$ and carry an $H$-action. Our definition unifies and generalises several familiar notions: taking $\chi = id: G \to G$ with the conjugation action recovers $G$-crossed braided categories; taking $\chi: G \to \{*\}$ for abelian $G$ yields $G$-graded braided categories; taking $\chi: \{*\} \to G$ leads to braided categories equipped with a $G$-action. The equivalence relation between $\chi$-crossed braided categories is typically finer than that between $G$-crossed braided ones. We classify $\chi$-crossed braided structures on the category of $G$-graded vector spaces in terms of cohomological data, and give explicit examples for cyclic groups. Given a doubly central algebra with $G$- and $H$-actions in a braided monoidal category, we define a notion of twisted-local modules and show how they give rise to $\chi$-crossed braided categories. We furthermore give sufficient conditions so that these categories are additionally $\chi$-crossed ribbon or admit an orthogonal $G$-decomposition.

We explore some methods to obtain cotorsion pairs and model category structures from a Frobenius pair. Specifically, we consider a type of Frobenius pair $(\mathcal{X},\omega)$ in an abelian category $\mathfrak{C}$ for which there exists a nonnegative integer $m$ such that every object in the right orthogonal complement $\mathcal{X}^{\perp_1}$ of $\mathcal{X}$ under $\text{Ext}^1_{\mathfrak{C}}(-,\sim)$ has a special $\omega^\wedge_m$-precover and a special $[\omega^\wedge_m]^{\perp_1}$-preenvelope, where $\omega^\wedge_m$ denotes the class of objects in $\mathfrak{C}$ with resolution dimension relative to $\omega$ at most $m$. The cotorsion pairs and model structures obtained from these Frobenius pairs will involve the classes $\mathcal{X}$, $\omega$, $\mathcal{X}^\wedge_m$, $\omega^\wedge_m$, and their orthogonal complements, in the description of its cofibrant, fibrant and trivial objects. As applications of our results, we obtain several cotorsion pairs and model structures related to relative and absolute Gorenstein homological dimensions.

Building on the dagger Bergner and dagger Joyal models constructed in Part I, we develop simplicial-space models for dagger $(\infty,1)$-categories. We first construct a homotopically constant resolution model and a model structure for unitarily complete dagger Segal spaces. A change-of-index adjunction compares the two presentations of dagger simplicial spaces. We then prove a dagger analogue of the Joyal–Tierney Quillen equivalence. Together with Part I, these results give a chain of Quillen equivalences between the dagger Bergner, dagger Joyal, and dagger Rezk models.

We study bounded derived categories of semisimple tensor categories and their semisimple module categories. These categories may have infinitely many simple isomorphism classes. We classify bounded monoidal $t$-structures by integer-valued characters of the universal grading group and an integer deviation. We also classify compatible module $t$-structures using stabilizer subgroups of the module components. The corresponding hearts are tensor and module equivalent to the original categories. It follows that a monoidal derived equivalence and a coherent derived module equivalence recover both the tensor category and its module category.

We prove a Snake Lemma for 2-categories. Working in a 2-category with a strong bizero object, we develop 2-kernels and 2-cokernels, 2-monomorphisms and 2-epimorphisms as fully faithful and cofully faithful 1-cells, short 2-exact sequences and normal image factorisations, and we prove a two-dimensional Normal Short Five Lemma. We then introduce the dinversion of an antinormal pair and the notion of a homologically self-dual 2-category, characterised equally by the self-duality of homology, by a Pure Snake Lemma and by a Third Isomorphism Property. Two-dimensional di-exactness implies homological self-duality. Our main result is the Snake Lemma in a 2-di-exact 2-category: a ladder of 2-exact rows with normal verticals induces a 2-exact six-term sequence, with a connecting 1-cell that is 2-natural in the ladder. Di-exactness can be traded for two hypotheses that are not self-dual: that dinversion preserve normality, and that normal 2-epimorphisms compose. Everything specialises, on passing to a locally discrete 2-category, to its classical counterpart. We close by exhibiting three models: a 2-di-exact 2-category of abelian categories containing $\mathsf{Coh}(X)$ for every noetherian scheme $X$; the locally ordered 2-category of complete modular lattices, in which 2-di-exactness amounts to Dedekind's transposition principle; and the 2-category of Hilbert lattices, which is not 2-di-exact but satisfies the non-self-dual hypotheses, by a theorem of Mackey on pairs of closed subspaces.

We introduce a notion of $2$-homological category modelled on pointed homological categories in the sense of Grandis, using bizero objects whose null $1$-cells are zero objects in the hom-categories for formulating $2$-dimensional pointedness. We first prove directly that the $2$-category of di-exact homological categories, which generalize Puppe exact categories, functors preserving all kernels and cokernels, and arbitrary natural transformations is $2$-homological. Its normal subcategories are saturated thick subcategories, and its exact quotients are constructed by a complete three-arrow fraction calculus similar to the one known for Puppe exact categories. We then adapt this proof to prove that the $2$-category of triangulated categories is also $2$-homological; here, instead of the ternary fractions we use the well-known Verdier fractions. Abstracting the common quotient structure of these proofs yields a general criterion, using which we further establish that the $2$-categories of pointed, additive and abelian categories are also $2$-homological. The criterion also applies to categories enriched in semimodules over a fixed commutative rig, with a zero object, and to their full sub-$2$-category with finite biproducts. Their quotients are linear congruence quotients. Combining enrichment with Puppe exactness gives further examples whose quotients are exact linear localizations, including linear abelian categories as the finite-biproduct case. In the abelian and triangulated cases the normal subcategories and quotients are, respectively, Serre subcategories and Serre quotients, and thick triangulated subcategories and Verdier localizations.

We characterize the arithmetical categories - the exact Mal'tsev categories whose lattices of congruences are distributive - by a simple condition on some suitable pushouts and a finite limit: a regular category is exact arithmetical if and only if, for every triple of regular epimorphisms with common domain, the pairwise pushouts exist and the comparison morphism to the limit of the diagram they form is a regular epimorphism. For a regular category, the validity of this condition for pairs and for triples is equivalent to its validity for $n$-tuples for all $n \ge 2$, and characterizes arithmetical categories among regular ones. For pairs alone the condition reduces to the characterization of exact Mal'tsev categories among regular ones, due to Carboni, Kelly and Pedicchio. For $n$-tuples with $n \ge 4$ nothing further is obtained: arithmeticity is the third and last rung of that ladder. The proof rests on a categorical form of the Chinese Remainder Theorem; in the exact Mal'tsev context this form of the theorem lies close to results of Hoefnagel on majority categories. As a direct application, we give a new proof of Pixley's characterization of arithmetical varieties using a suitable diagram of free algebras.

We construct the Cartan calculus of differential cubical forms on an object $X$ of a cartesian tangent category with a scalar multiplication by a commutative ring object $R$. We prove that the differential cubical forms are naturally equipped with the structure of a commutative differential graded algebra. Then we show that, if $R$ has no $2$-torsion, this CDGA together with the Lie algebra of vector fields and the inner derivatives constitutes a Cartan calculus, which lies between the initial Cartan calculus of algebraic Kaehler forms and the terminal one of Lie-Rinehart forms. We give a number of examples, including affine schemes and elastic diffeological spaces, and show that we recover the established notions of differential forms in all cases.

Let $[0,1]_*$ be the unit interval $[0,1]$ equipped with a left-continuous t-norm $*$. We prove that $[0,1]_*\text{-}\mathbf{Set}$ is cartesian closed if and only if $*$ is the minimum t-norm. This extends the classification for continuous t-norms established in the first part of this work to the whole left-continuous setting.

This article develops the notion of a computational category, motivated by the observation that many mathematical structures occurring in scientific computing already admit concrete algorithmic representations. Finite sets, finite-dimensional vector spaces, relations, and graphs may all be represented through elementary MATLAB and Octave data structures such as indexing vectors, matrices, logical arrays, and sparse matrices. The central idea is that a category may often be described through a category of canonical representatives indexed by the natural numbers. This viewpoint leads naturally to representative categories whose objects are dimensions and whose morphisms are finite computational data structures. The resulting framework unifies several examples, including matrix categories, index categories, categories of column spaces, row-based models of finite sets, and categories combining finite subsets with finite-dimensional vector spaces through mixed morphisms. The paper also introduces computational categories with distinguished non-finite representatives, extending the finite setting while retaining concrete coordinate descriptions. Throughout, categorical constructions are interpreted algorithmically and implemented using standard MATLAB and Octave operations. The work contributes to a computational perspective on category theory and, conversely, to a categorical understanding of matrix-oriented scientific computing.

We construct a weak Lie 3-group $T_2 \mathbb B^{F_1}_n$ from the 2-category of gerbes over $\mathbb R^n / \mathbb Z^n$ and the $\mathbb R^n/ \mathbb Z^n$-action on it by pullback along translations. We also construct a homotopy equivalence between $T_2 \mathbb B^{F_1}_n$ and a different Lie 3-group $T_2 \mathbb D^{F_1}_n$, which admits a dimensional reduction to the homotopy equivalence of Lie 2-groups introduced by Nikolaus and Waldorf to model half-geometric T-duality. This is a first step towards establishing a higher form of T-duality for 2-gerbes over torus fibrations, relevant to supergravity and M-theory.